Estimation of Homography Dynamics on the Special Linear Group - Visual Servoing via Advanced Numerical Methods - page 131

the estimation error H = H − 1 H to the identity matrix I , and the estimation error
A = A

− A to zero.
The estimator filter equation of H is posed directly on SL (3). It includes a cor-
rection term derived from the error H . We consider an estimator filter of the form
˙

H = H Ad H A +

H ) ( H

H (0)= H 0 ,

α , ,

(8.5)

˙

A = A (0)= A 0 .

( H β , H ) ,

This yields the following expression for the dynamics of the estimation error
( H , A )=( H − 1 H − A ): , A

H = H

˙

A

− Ad H − 1 α

(8.6)

˙

A =

− β

with the arguments of α and β omitted to lighten the notation. The main result of

the chapter is stated next.

Theorem 8.1. Assume that the matrix A in (8.4) is constant. Consider the nonlinear
estimator filter (8.5) along with the innovation α and the estimation dynamics β

defined as

⎧
⎨

( H T ( I − H ))

α = − k H Ad H P , k H > 0

(8.7)

⎩

( H T ( I − H )) β = − k A P , k A > 0

3 × 3

with the projection operator P : R → sl (3) defined by (8.2) . Then, for the esti-

mation error dynamics (8.6) ,

∪

i) all solutions converge to E = E s E u with:

E s =( I , 0)

( H 0 , | H 0 =

λ − 3 1) vv )

2

E u = { 0) λ ( I +( − , v ∈ S }

3 2 + 1 = 0 ;

where λ ≈− 0 . 7549 is the unique real solution of the equation λ − λ

0) is locally exponentially stable;
iii) any point of E u is an unstable equilibrium. More precisely, for any ( H 0 ,

ii) the equilibrium point E s =( I ,

0) ∈ E u

0) , there exists ( H 1 , A 1 )

of ( H 0 , and any neighborhood U ∈ U such that the

solution of system (8.6) issued from ( H 1 , A 1 ) converges to E s .

Proof. Let us prove part i) . Let us consider the following candidate Lyapunov
function

, A )= 1

1

V ( H − H

2 k A A

2 + 2

2

I

(8.8)

= 1

1
2 k A tr( A T

− H ) T ( I − H )) +

A )

2 tr(( I

.